Time-dependent radial wavefunction

Fig. 2. Schematic configuration space for the reaction AB + CD — A + BCD. R is the radial coordinate between the center-of-mass of the two diatoms, and r is the vibrational coordinate of the reactive AB diatom. I denotes the interaction region and II denotes the asymptotic region. The shaded regions are the absorption zones for the time-dependent wavefunction to avoid boundary reflections. The reactive flux is evaluated at the r = rB surface.

We begin our discussion with the simple case of a spinless particle of mass m and kinetic energy E = h2k2/2m in a spherical, time-independent potential V(r), so that the Schrodinger equation can be decomposed into uncoupled partial waves l. For a particular l, the scattering matrix or the S matrix is defined as S(k) = exp[2/5(A )] in terms of the phase shift 8(k). Here and in the following, the subscript l on the S matrix and the phase shift is suppressed. The asymptotic form of the time-dependent radial wavefunction is expressible as... 

We show how one can image the amplitude and phase of bound, quasibound and continuum wavefunctions, using time-resolved and frequency-resolved fluorescence. The case of unpolarized rotating molecules is considered. Explicit formulae for the extraction of the angular and radial dependence of the excited-state wavepackets are developed. The procedure is demonstrated in Na2 for excited-state wavepackets created by ultra-short pulse excitations. 

The wavefunction can be used to obtain the probability of finding an electron at any particular distance from the nucleus by integrating over the surface of the sphere at that distance. As can be seen from Figure 7.12, the infinitesimal volume concerned depends on the surface area of the sphere of radius r. The mathematics of this is discussed in more detail in Appendix 9, but here we note that the relevant function for the radial probability is actually Anr R rf. The Anr factor arises because the infinitesimal volume at a particular distance from the nuclear centre increases with So there is four times more volume at 2 A from the nucleus than at 1 A. The plots of the probability function for Is, 2s and 2p orbitals are shown in Figure 7.1 lb. 

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